Example 37.3 Maximum Likelihood Factor Analysis

This example uses maximum likelihood factor analyses for one, two, and three factors. It is already apparent from the principal
factor analysis that the best number of common factors is almost certainly two. The one- and three-factor ML solutions reinforce
this conclusion and illustrate some of the numerical problems that can occur. The following statements produce Output 37.3.1 through Output 37.3.3:

Eigenvalues of the Weighted ReducedCorrelation Matrix: Total = 0 Average = 0

Eigenvalue

Difference

1

Infty

Infty

2

1.92716032

2.15547340

3

-.22831308

0.56464322

4

-.79295630

0.11293464

5

-.90589094

Factor Pattern

Factor1

Population

0.97245

School

0.15428

Employment

1.00000

Services

0.51472

HouseValue

0.12193

Variance Explained by Each Factor

Factor

Weighted

Unweighted

Factor1

17.8010629

2.24926004

Final Communality Estimates and VariableWeights

Total Communality: Weighted = 17.801063 Unweighted = 2.249260

Variable

Communality

Weight

Population

0.94565561

18.4011648

School

0.02380349

1.0243839

Employment

1.00000000

Infty

Services

0.26493499

1.3604239

HouseValue

0.01486595

1.0150903

The solution on the second iteration is so close to the optimum that PROC FACTOR cannot find a better solution; hence you
receive this message:

Convergence criterion satisfied.

When this message appears, you should try rerunning PROC FACTOR with different prior communality estimates to make sure that
the solution is correct. In this case, other prior estimates lead to the same solution or possibly to worse local optima,
as indicated by the information criteria or the chi-square values.

The variable Employment has a communality of 1.0 and, therefore, an infinite weight that is displayed next to the final communality estimate as a
missing/infinite value. The first eigenvalue is also infinite. Infinite values are ignored in computing the total of the eigenvalues
and the total final communality.

Output 37.3.2 displays the results of the analysis with two factors. The analysis converges without incident. This time, however, the Population variable is a Heywood case.

Output 37.3.2: Maximum Likelihood Factor Analysis: Two Factors

Input Data Type

Raw Data

Number of Records Read

12

Number of Records Used

12

N for Significance Tests

12

Prior Communality Estimates: SMC

Population

School

Employment

Services

HouseValue

0.96859160

0.82228514

0.96918082

0.78572440

0.84701921

Preliminary Eigenvalues: Total = 76.1165859 Average = 15.2233172

Eigenvalue

Difference

Proportion

Cumulative

1

63.7010086

50.6462895

0.8369

0.8369

2

13.0547191

12.7270798

0.1715

1.0084

3

0.3276393

0.6749199

0.0043

1.0127

4

-0.3472805

0.2722202

-0.0046

1.0081

5

-0.6195007

-0.0081

1.0000

2 factors will be retained by the NFACTOR criterion.

Iteration

Criterion

Ridge

Change

Communalities

1

0.3431221

0.0000

0.0471

1.00000

0.80672

0.95058

0.79348

0.89412

2

0.3072178

0.0000

0.0307

1.00000

0.80821

0.96023

0.81048

0.92480

3

0.3067860

0.0000

0.0063

1.00000

0.81149

0.95948

0.81677

0.92023

4

0.3067373

0.0000

0.0022

1.00000

0.80985

0.95963

0.81498

0.92241

5

0.3067321

0.0000

0.0007

1.00000

0.81019

0.95955

0.81569

0.92187

Convergence criterion satisfied.

Significance Tests Based on 12 Observations

Test

DF

Chi-Square

Pr > ChiSq

H0: No common factors

10

54.2517

<.0001

HA: At least one common factor

H0: 2 Factors are sufficient

1

2.1982

0.1382

HA: More factors are needed

Chi-Square without Bartlett's Correction

3.3740530

Akaike's Information Criterion

1.3740530

Schwarz's Bayesian Criterion

0.8891463

Tucker and Lewis's Reliability Coefficient

0.7292200

Squared Canonical Correlations

Factor1

Factor2

1.0000000

0.9518891

Eigenvalues of the Weighted Reduced Correlation Matrix: Total = 19.7853157 Average = 4.94632893

Eigenvalues of the Weighted Reduced Correlation Matrix: Total = 41.5254193 Average = 10.3813548

Eigenvalue

Difference

Proportion

Cumulative

1

Infty

Infty

2

39.3054826

37.0854258

0.9465

0.9465

3

2.2200568

2.2199693

0.0535

1.0000

4

0.0000875

0.0002949

0.0000

1.0000

5

-0.0002075

-0.0000

1.0000

Factor Pattern

Factor1

Factor2

Factor3

Population

0.97245

-0.11233

-0.15409

School

0.15428

0.89108

0.26083

Employment

1.00000

0.00000

0.00000

Services

0.51472

0.72416

-0.12766

HouseValue

0.12193

0.97227

-0.08473

Variance Explained by Each Factor

Factor

Weighted

Unweighted

Factor1

54.6115241

2.24926004

Factor2

39.3054826

2.27634375

Factor3

2.2200568

0.11525433

Final Communality Estimates and VariableWeights

Total Communality: Weighted = 96.137063 Unweighted = 4.640858

Variable

Communality

Weight

Population

0.98201660

55.6066901

School

0.88585165

8.7607194

Employment

1.00000000

Infty

Services

0.80564301

5.1444261

HouseValue

0.96734687

30.6251078

In the results, a warning message is displayed:

WARNING: Too many factors for a unique solution.

The number of parameters in the model exceeds the number of elements in the correlation matrix from which they can be estimated,
so an infinite number of different perfect solutions can be obtained. The criterion approaches zero at an improper optimum,
as indicated by this message:

Converged, but not to a proper optimum.

The degrees of freedom for the chi-square test are –2, so a probability level cannot be computed for three factors. Note also
that the variable Employment is a Heywood case again.

The probability levels for the chi-square test are 0.0001 for the hypothesis of no common factors, 0.0002 for one common factor,
and 0.1382 for two common factors. Therefore, the two-factor model seems to be an adequate representation. Akaike’s information
criterion and Schwarz’s Bayesian criterion attain their minimum values at two common factors, so there is little doubt that
two factors are appropriate for these data.